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Properties

Label 624.2.u.c
Level $624$
Weight $2$
Character orbit 624.u
Analytic conductor $4.983$
Analytic rank $0$
Dimension $208$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [624,2,Mod(5,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.u (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.98266508613\)
Analytic rank: \(0\)
Dimension: \(208\)
Relative dimension: \(104\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 208 q - 2 q^{3} + 4 q^{6} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 208 q - 2 q^{3} + 4 q^{6} - 12 q^{7} + 16 q^{10} - 4 q^{12} + 20 q^{13} - 6 q^{15} + 24 q^{16} + 8 q^{18} - 8 q^{22} - 12 q^{24} - 216 q^{25} - 20 q^{27} - 8 q^{28} + 24 q^{30} - 24 q^{31} - 28 q^{33} - 16 q^{34} - 52 q^{36} - 10 q^{39} + 16 q^{40} + 4 q^{43} + 32 q^{46} - 32 q^{48} - 18 q^{51} + 8 q^{52} + 44 q^{54} + 32 q^{57} - 88 q^{58} + 24 q^{61} + 46 q^{63} + 48 q^{64} + 16 q^{66} + 8 q^{69} - 48 q^{70} - 56 q^{72} - 48 q^{73} + 28 q^{75} + 64 q^{76} - 44 q^{78} + 48 q^{79} - 36 q^{81} - 8 q^{82} - 12 q^{84} - 80 q^{88} + 12 q^{90} + 4 q^{91} - 32 q^{94} - 44 q^{96} + 56 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −1.41421 − 0.00417789i −1.19223 + 1.25642i 1.99997 + 0.0118168i − 3.65281i 1.69131 − 1.77186i −0.697387 + 0.697387i −2.82832 − 0.0250670i −0.157179 − 2.99588i −0.0152610 + 5.16583i
5.2 −1.41420 + 0.00500305i 1.45866 − 0.933975i 1.99995 − 0.0141507i − 0.894249i −2.05817 + 1.32813i 2.54390 − 2.54390i −2.82827 + 0.0300178i 1.25538 − 2.72471i 0.00447398 + 1.26465i
5.3 −1.41304 + 0.0575842i 0.353518 + 1.69559i 1.99337 − 0.162738i 0.498043i −0.597174 − 2.37558i 1.48729 − 1.48729i −2.80734 + 0.344742i −2.75005 + 1.19884i −0.0286794 − 0.703754i
5.4 −1.40979 + 0.111747i 1.70866 + 0.283700i 1.97503 − 0.315079i 4.30926i −2.44056 − 0.209021i −2.57188 + 2.57188i −2.74917 + 0.664898i 2.83903 + 0.969493i −0.481544 − 6.07515i
5.5 −1.40143 − 0.189719i 0.611115 − 1.62066i 1.92801 + 0.531755i 2.12698i −1.16390 + 2.15530i −0.901700 + 0.901700i −2.60109 − 1.11100i −2.25308 − 1.98082i 0.403528 − 2.98082i
5.6 −1.39393 + 0.238653i −1.62668 − 0.594915i 1.88609 − 0.665333i − 2.52745i 2.40945 + 0.441058i 1.71934 − 1.71934i −2.47029 + 1.37755i 2.29215 + 1.93547i 0.603184 + 3.52309i
5.7 −1.38615 − 0.280345i −1.30843 − 1.13491i 1.84281 + 0.777199i 0.702388i 1.49551 + 1.93996i −2.90271 + 2.90271i −2.33653 − 1.59394i 0.423966 + 2.96989i 0.196911 − 0.973613i
5.8 −1.37571 + 0.327765i 1.57603 − 0.718425i 1.78514 − 0.901816i − 1.45205i −1.93268 + 1.50491i −1.35916 + 1.35916i −2.16025 + 1.82574i 1.96773 − 2.26452i 0.475930 + 1.99759i
5.9 −1.37360 − 0.336494i 1.04018 + 1.38493i 1.77354 + 0.924416i 0.317135i −0.962764 − 2.25235i −0.114536 + 0.114536i −2.12507 − 1.86656i −0.836064 + 2.88115i 0.106714 − 0.435617i
5.10 −1.36708 − 0.362058i −1.50124 + 0.863869i 1.73783 + 0.989925i 2.89815i 2.36509 − 0.637444i 2.47586 − 2.47586i −2.01734 − 1.98250i 1.50746 − 2.59376i 1.04930 − 3.96202i
5.11 −1.36184 − 0.381305i 1.64467 + 0.543198i 1.70921 + 1.03855i − 3.81978i −2.03265 − 1.36687i −0.258292 + 0.258292i −1.93167 − 2.06607i 2.40987 + 1.78676i −1.45650 + 5.20193i
5.12 −1.34216 + 0.445665i −1.30358 − 1.14047i 1.60277 − 1.19630i 3.31873i 2.25788 + 0.949734i 1.02244 − 1.02244i −1.61801 + 2.31992i 0.398641 + 2.97340i −1.47904 − 4.45426i
5.13 −1.34007 − 0.451906i −1.44909 + 0.948758i 1.59156 + 1.21117i 0.526414i 2.37063 − 0.616547i −2.14900 + 2.14900i −1.58547 − 2.34229i 1.19972 − 2.74967i 0.237890 − 0.705430i
5.14 −1.33816 + 0.457521i −0.180369 − 1.72263i 1.58135 − 1.22447i − 3.10319i 1.02950 + 2.22264i −2.19073 + 2.19073i −1.55588 + 2.36204i −2.93493 + 0.621418i 1.41978 + 4.15257i
5.15 −1.30537 + 0.544058i −0.860631 + 1.50310i 1.40800 − 1.42040i − 0.632967i 0.305672 − 2.43034i 0.779558 − 0.779558i −1.06519 + 2.62018i −1.51863 − 2.58723i 0.344371 + 0.826259i
5.16 −1.26632 + 0.629625i 1.54690 + 0.779159i 1.20715 − 1.59462i 1.77288i −2.44946 − 0.0126977i 2.46507 − 2.46507i −0.524627 + 2.77935i 1.78582 + 2.41057i −1.11625 − 2.24504i
5.17 −1.21467 + 0.724270i −0.468874 + 1.66738i 0.950865 − 1.75950i 2.93002i −0.638105 − 2.36491i −1.92974 + 1.92974i 0.119367 + 2.82591i −2.56031 − 1.56358i −2.12213 − 3.55902i
5.18 −1.21418 − 0.725097i −0.247322 − 1.71430i 0.948468 + 1.76080i − 2.53765i −0.942742 + 2.26080i 2.19282 − 2.19282i 0.125138 − 2.82566i −2.87766 + 0.847969i −1.84004 + 3.08116i
5.19 −1.17332 + 0.789508i 0.910598 + 1.47337i 0.753353 − 1.85269i − 2.78774i −2.23166 − 1.00980i −2.76731 + 2.76731i 0.578791 + 2.76857i −1.34162 + 2.68329i 2.20094 + 3.27091i
5.20 −1.16346 − 0.803971i −1.60863 − 0.642106i 0.707263 + 1.87077i 0.349564i 1.35534 + 2.04036i 1.86313 − 1.86313i 0.681174 − 2.74518i 2.17540 + 2.06583i 0.281039 − 0.406703i
See next 80 embeddings (of 208 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.104
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
208.r odd 4 1 inner
624.u even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.u.c ✓ 208
3.b odd 2 1 inner 624.2.u.c ✓ 208
13.d odd 4 1 624.2.bm.c yes 208
16.e even 4 1 624.2.bm.c yes 208
39.f even 4 1 624.2.bm.c yes 208
48.i odd 4 1 624.2.bm.c yes 208
208.r odd 4 1 inner 624.2.u.c ✓ 208
624.u even 4 1 inner 624.2.u.c ✓ 208
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.2.u.c ✓ 208 1.a even 1 1 trivial
624.2.u.c ✓ 208 3.b odd 2 1 inner
624.2.u.c ✓ 208 208.r odd 4 1 inner
624.2.u.c ✓ 208 624.u even 4 1 inner
624.2.bm.c yes 208 13.d odd 4 1
624.2.bm.c yes 208 16.e even 4 1
624.2.bm.c yes 208 39.f even 4 1
624.2.bm.c yes 208 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\):

\( T_{5}^{104} + 314 T_{5}^{102} + 47587 T_{5}^{100} + 4637600 T_{5}^{98} + 326709961 T_{5}^{96} + \cdots + 71\!\cdots\!56 \) Copy content Toggle raw display
\( T_{11}^{104} - 604 T_{11}^{102} + 175628 T_{11}^{100} - 32751408 T_{11}^{98} + 4402522160 T_{11}^{96} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display